# Illustration of Von Neumann's Minimax theorem in games?

The Von Neumann's Minimax theorem gives the conditions that make the max-min inequality an equality.

I understand the max-min inequality, basically min(max(f))>=max(min(f)).

The Von Neumann's theorem states that, for the inequality to become an equality f(.,y) should always be convex for given y and f(x,.) should always be concave for given x, which also makes sense.

This video says that for a zero-sum perfect information game, the Von Neumann's theorem always holds, so that minimax always equal to maximin, which I did not quite follow.

Questions
Why zero-sum perfect information games satisfy the conditions of Von Neumann's theorem?
If we relax the rules to be non-zero-sum or non-perfect information, how would the conditions change?

• Why won't you accept the edit? See this post on meta. And please stop rolling it back. – Mithical Aug 11 '16 at 14:38
• @Mithrandir see my answer on that post, at least the game-theory tag should be there – dontloo Aug 11 '16 at 14:39
• I'll agree with the difference between game-theory and gaming. But can you please edit the other tags out? – Mithical Aug 11 '16 at 14:40
• @Mithrandir sure – dontloo Aug 11 '16 at 14:43
• This is an interesting questions - you've clearly done your research - but unfortunately, it seems better for Cross Validated or Data Science. For more information, visit Artificial Intelligence Meta. – Ben N Aug 16 '16 at 16:01

Here is one of many web pages devoted to the application of Minimax in Tic-Tac-Toe solving: Minimax Algorithm in Game Theory | Set 3 (Tic-Tac-Toe AI – Finding optimal move)